Carl Smart is currently (at time t = 0) considering a couple of investment projects that will provide him with a dividend in one year from now (time t = 1) and a dividend in two years from now (time t = 2). He figures that the size of the dividends will depend on the growth rate of aggregate consumption over these two years. The first project Carl considers provides a dividend of Dt = 60t + 5(Ct − E[Ct])

at t = 1 and at t = 2. The second project provides a dividend of Dt = 60t − 5(Ct − E[Ct])

at t = 1 and at t = 2. Here E[ ] is the expectation computed at time 0 and Ct denotes aggregate consumption at time t. The current level of aggregate consumption is C0 = 1000.

As a valuable input to his investment decision Carl wants to compute the present value of the future dividends of each of the two projects.

First, Carl computes the present values of the two projects using a ‘risk-ignoring approach’, that is by discounting the expected dividends using the riskless returns observed in the bond markets. Carl observes that a one-year zero-coupon bond with a face value of 1000 currently trades at a price of 960 and a two-year zero-coupon bond with a face value of 1000 trades at a price of 929.02.

(a) What are the expected dividends of project 1 and project 2?

(b) What are the present values of project 1 and project 2 using the risk-ignoring approach?
Suddenly, Carl remembers that he once took a great course on advanced asset pricing and that the present value of an uncertain dividend of D1 at time 1 and an uncertain dividend of D2 at time 2 should be computed as P = E 
ζ1 ζ0 D1 + ζ2 ζ0 D2 
, where the ζt’s define a state-price deflator. After some reflection and analysis, Carl decides to value the projects using a conditional consumption-based CAPM so that the state-price deflator between time t and t + 1 is of the form ζt+1 ζt = at + bt Ct+1 Ct , t = 0, 1, ... .
Carl thinks it’s fair to assume that aggregate consumption will grow by either 1% (‘low’) or 3% (‘high’) in each of the next two years. Over the first year he believes the two growth rates are equally likely. If the growth rate of aggregate consumption is high in the first year, he believes that there is a 30% chance that it will also be high in the second year and, thus, a 70% chance of low growth in the second year. On the other hand, if the growth rate of aggregate consumption is low in the first year, he estimates that there will be a 70% chance of high growth and a 30% chance of low growth in the second year.
In order to apply the consumption-based CAPM for valuing his investment projects, Carl has to identify the coefficients at and bt fort = 0, 1.The values of a0 and b0 can be identified from the prices of two traded assets that only provide dividends at time 1. In addition to the one-year zero-coupon bond mentioned above, a one-year European call option on aggregate consumption is traded. The option has a strike price of K = 1020 so that the payoff of the option in one year is max(C1 − 1020, 0), where C1 is the aggregate consumption level at t = 1. The option trades at a price of 4.7.

(c) Using the information on the two traded assets, write up two equations that can be used for determining a0 and b0. Verify that the equations are solved for a0 = 3 and b0 = −2.
The values of a1 and b1 may depend on the consumption growth rate of the first year, that is Carl has to find ah 1, bh 1 that defines the second-year state-price deflator if the first-year growth rate was high and al 1, bl 1 that defines the second-year state-price deflator if the firstyear growth rate was low. Using the observed market prices of other assets, Carl concludes that ah 1 = 2.5, bh 1 = −1.5, al 1 = 3.5198, bl 1 = −2.5.

(d) Verify that the economy is path-independent in the sense that the current price of an asset that will pay you 1 at t = 2 if the growth rate is high in the first year and low in the second year will be identical (at least to five decimal places) to the current price of an asset that will pay you 1 at t = 2 if the growth rate is low in the first year and high in the second year.

(e) Illustrate the possible dividends of the two projects in a two-period binomial tree.

(f) What are the correctly computed present values of the two projects?
(g) Carl notes that the expected dividends of the two projects are exactly the same but the present value of project 2 is higher than the present value of project 1. Although Carl is pretty smart, he cannot really figure out why this is so. Can you explain it to him?